where \(\hat{D}\)  is the surfactant bulk diffusivity.
The interfacial surfactant is coupled to the flow by a steady advection–diffusion equation and an equation of state.
The bulk concentration is coupled to the interfacial surfactant by continuity of flux, where exchange at the interface is modelled using a source–sink term consistent with the Henry isotherm [1]}.
The equation of state and adsorption–desorption kinetics are linearised, which is valid for small deviations in the concentration of surfactant away from some reference value [2]}.
On \(\hat{\mathcal {I}}\) , we balance the tangential components of the stress, \(\hat{{T}\,}\cdot n\)  where \(\hat{{T}\,} = -\hat{p} {I} + \hat{\mu }(\hat{\nabla }\hat{u} + (\hat{\nabla }\hat{u})^{T})\)  and \(n\)  is the unit normal to \(\hat{\mathcal {I}}\)  (pointing into the channel), with tangential gradients of surface tension \(\hat{\sigma }\) .
We assume that (i) the effects arising from the gas trapped in the SHSs are negligible, (ii) \(\hat{\sigma } = \hat{\sigma }_0 - \hat{A}(\hat{\Gamma } - \hat{\Gamma }_0)\)  where \(\hat{\sigma }_0\)  is the reference surface tension, \(\hat{A}\)  is the surface activity and \(\hat{\Gamma }_0\)  is the reference surface concentration of surfactant, and (iii) the surface tension remains large enough to suppress deflections of the interface from its assumed flat state.
Thus, along \(\hat{\mathcal {I}}\) , we impose the tangential stress balance in the streamwise and transverse directions, no-penetration of velocity, continuity of surfactant flux and the transport equation for interfacial surfactant
\(\hat{\mu } n\cdot \hat{\nabla } \hat{u} - \hat{A} \hat{\Gamma }_{\hat{x}} = 0, \quad \hat{\mu } n\cdot \hat{\nabla } \hat{w} - \hat{A} \hat{\Gamma }_{\hat{z}} = 0, \quad \hat{v}=0, \quad \hat{D} n\cdot \hat{\nabla } \hat{c} - \hat{K}_a \hat{c} + \hat{K}_d \hat{\Gamma } =0, \\ \quad \hat{D}_I ( \hat{\Gamma }_{\hat{x}\hat{x}} + \hat{\Gamma }_{\hat{z}\hat{z}} ) + \hat{K}_a \hat{c} - \hat{K}_d \hat{\Gamma } -(\hat{u} \hat{\Gamma })_{\hat{x}} - (\hat{w} \hat{\Gamma })_{\hat{z}} = 0,\qquad \mathrm {(2.6\textit {a--e})}\) 
